A Spatio-temporal Optimal, Hybird Compact–WENO Scheme with Minimized Dispersion and Critical-adaptive Dissipation for Solving Compressible Flows

In this paper, based on our previous optimal compact schemes with minimized dispersion and controllable dissipation (OC–WENO schemes) (Sun et al. in Sci China-Phys Mech Astron 57:971–982, 2014), a spatio-temporal optimized, hybird compact-WENO scheme with minimized dispersion and critical-adaptive dissipation is developed for solving compressible flows. Firstly, the spectral properties of the fourth-order OC–WENO scheme is researched within the spatio-temporal discrete framework. In conjunction with total dispersion error of the fully scheme, an integrated error function is designed to optimize the dispersion property. Secondly, the scale sensor leveraged to quantify the local scaled wavenumber is optimized in the wavenumber space to improve the accuracy of estimating. Moreover, a dispersion-dissipation condition, controlling the relative proportion of dispersion and dissipation errors, is developed for the fully discrete scheme. Thirdly, by exploiting the optimized scale sensor and the dispersion-dissipation condition, the critical-adaptive dissipation surface is constructed to achieve the adaptive dissipation property relating to the local characteristics of the flow fields and different Courant number. To have the shock-capturing capability, the proposed compact scheme is blended with fifth-order WENO scheme to form the MDADFC–WENO scheme. Finally, a set of benchmark test cases is employed to validate the good performance of the proposed scheme.